Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems

Abstract

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method.

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

“…Many authors established and generalized several results on the existence and nature of the solution of the equilibrium problems (see for more detail [1,4,5]). Due to the importance of this problem (EP) in both pure and applied sciences, many researchers studied it in recent years [6][7][8][9][10][11][12][13][14][15][16][17] and other in [18][19][20][21][22]. Tran et al in [23] introduced iterative sequence {u n } in the following way:…”

Halpern subgradient extragradient algorithm for solving quasimonotone variational inequality problems

“…The first well-known projection method is the gradient projection method that is used to solve variational inequalities. Moreover, several other projection methods have been established including the well-known extragradient method [18] the subgradient extragradient method [4,5] and others in [6,26,20,30,11] and others in [22,7,23,14,9,27,28,2,1,10]. The above numerical techniques are used to examine the variational inequalities involving monotone, strongly monotone, or inverse monotone.…”

Bounded perturbation resilience of viscosity proximal algorithm for solving split variational inclusion problems with applications to compressed sensing and image recovery

“…The SFP has real-life scenarios in signal processing, intensity-modulated radiation therapy (IMRT) treatment planning, sensor networks in computerized tomography, magnetic resonance imaging, graph matching, and so forth; see, for example, previous studies. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] Though there is "no one size fits all" iterative method for solving most nonlinear problems. However, for the SFP (3), there exist a proximal split algorithm due to Moudafi and Thakur 16 and the relaxed CQ algorithm with self adaptive step size of Kazmi and Rizvi 17 with nice convergence behavior.…”

“…The SFP has real‐life scenarios in signal processing, intensity‐modulated radiation therapy (IMRT) treatment planning, sensor networks in computerized tomography, magnetic resonance imaging, graph matching, and so forth; see, for example, previous studies 2‐15 …”